Optimal polynomial blow up range for critical wave maps

We prove that the critical Wave Maps equation with target $S^2$ and origin ℝ$^{2+1}$ admits energy class blow up solutions of the form \[ u(t, r) = Q(\lambda(t)r) + \epsilon(t, r) \] where $Q:ℝ²\rightarrow S^2$ is the ground state harmonic map and $\lambda(t) = t^{-1-\nu}$ for any $\nu>0$. This extends the work, where such solutions were constructed under the assumption $\nu>\frac{1}{2}$. Also in the later chapter, we give the necessary remarks and key changes one needs to notice while the same problem is considered in a more general case while $\cal{N}$ is a surface of revolution. We are also able to extends the blow-up range in Carstea's work to $\nu>0$. In light of a result of Struwe, our results are optimal for polynomial blow up rates.